PHYSICAL REVIEW B 88, 195201 (2013)
Defect physics and electronic properties of Cu3 PSe4 from first principles
D. H. Foster, F. L. Barras, J. M. Vielma, and G. Schneider*
Department of Physics, Oregon State University, Corvallis, Oregon 97331, USA
(Received 26 September 2012; revised manuscript received 6 September 2013; published 8 November 2013)
The p-type semiconductor Cu3 PSe4 has recently been established to have a direct band gap of 1.4 eV and
an optical absorption spectrum similar to GaAs [Foster et al., Appl. Phys. Lett. 99, 181903 (2011)], suggesting
a possible application as a solar photovoltaic absorber. Here we calculate the thermodynamic stability, defect
energies and concentrations, and several material properties of Cu3 PSe4 using a wholly GGA + U method (the
generalized gradient approximation of density functional theory with a Hubbard U term included for the Cu-d
orbitals). We find that two low energy acceptor defects, the copper vacancy VCu and the phosphorus-on-selenium
antisite PSe , establish the p-type behavior and likely prevent any n-type doping near thermal equilibrium. The
GGA + U defect calculation method is shown to yield more accurate results than the more standard method of
applying post-calculation GGA + U -based band-gap corrections to strictly GGA defect calculations.
DOI: 10.1103/PhysRevB.88.195201
PACS number(s): 61.72.J−, 71.20.Nr, 71.15.Mb, 88.40.fh
I. INTRODUCTION
The growing family of multinary copper chalcogenides
has been of great interest for solar photovoltaic applications. In addition to the commonly used solar absorber
CuIn1−x Gax Se2 (CIGS), materials that have raised interest
include Cu2 ZnSnS4 , Cu7 TlS4 ,1 CuClSe2 ,2 CuBiS2 ,3 CuSbS2 ,3
and Cu3 BiS3 .4 Recently the p-type semiconductor Cu3 PSe4
has been established5 to have a direct band gap of Eg =
1.4 eV, with a calculated absorption α > 5 × 104 cm−1 for
wavelengths less than 630 nm. This band gap lies in the
optimal range for photovoltaic power output and warrants
further investigation of the material.
In addition to optical absorption, essential considerations
for photovoltaic applications include ease of synthesis, conductivity, amenability to doping, and trap-assisted charge
recombination. These quantities are largely controlled by
the thermodynamic stability of the material with respect to
competing phases and point defects. Materials which allow
bipolar doping (both n-type and p-type behavior are achievable
through doping) are of special interest because p-n homojunction capability may reduce the number of heterojunctions
needed in a solar cell design. Bipolar doping occurs under
typical synthesis techniques only when all intrinsic defects
have charge transition energies and formation energies large
enough so that extrinsic (dopant) charged defect states are
energetically favorable for a sizable range of Fermi energies,
extending well above and below the center of the band gap.
Computational defect analysis using relatively inexpensive
methods can often determine with good confidence whether
bipolar doping is possible for synthesis methods near thermal
equilibrium.
Here we perform a point defect analysis of Cu3 PSe4
combining the + U Hubbard term for total energy calculations
with the correction methods described recently by Lany and
Zunger.6,7 Several potential substitutional donor defects are
also considered. Furthermore we examine bulk properties
including the partial density of states (DOS), the dielectric
tensor, and the highly asymmetric effective mass tensor. We
compare our results to recent experiments8 and to a more
standard procedure using the generalized gradient approximation (GGA) for defect supercell calculations followed by
1098-0121/2013/88(19)/195201(7)
a post-calculation valence band correction. We also compare
our methods with the alternative electrostatic image correction
procedure described by Freysoldt et al.9
II. METHODS
A. Computation
Defect formation energies are most often calculated using
density functional theory (DFT) within the local density
approximation (LDA) or within the GGA. However, recent
statistical studies10,11 on the accuracy of heat of formation
calculations indicate that using GGA with an additional
Hubbard U term for the occupation of transition metal d
orbitals, the so-called GGA + U method, will be more accurate
than using standard GGA or LDA. Furthermore, the defect
study by Scanlon et al.12 has compared the GGA and GGA + U
methods for VCu and IO defects in Cu2 O, and found that in
comparison to the GGA with a valence band correction, the
wholly GGA + U method reproduced more (although not all)
of the experimental features sought. In the GGA + U method,
the U value is held constant for each type of transition metal
atom throughout the analysis, including calculations of the
energies of the transition metal elements themselves.
The heat of formation studies10,11 also suggest that one
should add a statistically determined correction value to the
total energy of each pure element before calculating the heat
of formation H of a compound. To obtain the most accurate
heat of formation energies for both compounds and defects,
we use GGA + U 13 and apply the elemental energy corrections
suggested by Lany10 for P in all phosphides14 and for Ca in all
Ca compounds. The other elements we consider, Cu, Se, Zn,
Cd, and Cl, either have statistically insignificant corrections
or, in the case of Cl, are not considered in Ref. 10.
Our calculations use the projector augmented wave (PAW)
method15,16 as implemented in the plane wave code VASP17
with the Perdew-Burke-Ernzerhof18 (PBE) parametrization
of the GGA exchange-correlation functional. We use an
effective U value of 6 eV for the Cu-d, Zn-d, and Cd-d
orbitals. This value of U for Cu-d has been chosen in
previous work (c.f. Ref. 19) to yield agreement with the
experimental band structure below the valence band maximum
195201-1
©2013 American Physical Society
FOSTER, BARRAS, VIELMA, AND SCHNEIDER
PHYSICAL REVIEW B 88, 195201 (2013)
(VBM),20 thus eliminating or significantly reducing the need
for post-calculation corrections to the VBM of Cu3 PSe4 .21
Calculations use a plane wave cutoff energy of 310 eV and
a set of comparison calculations using cutoff energy 400 eV
resulted in very small corrections of order 0.01 eV. The density
functional perturbation theory calculations we report below
were calculated with a 400 eV cutoff. In the image charge
corrections and the hydrogenic binding energy estimations,
we have used the value 0 = 14.1, which was calculated with
the 310 eV cutoff. All calculations include ionic relaxation,
while lattice parameters are relaxed for all pure compounds and
elements, including the defect free host. Lattice parameters are
determined by performing shape relaxations for a sequence of
cell volumes, and interpolating the volume of minimum energy
using the Murnaghan equation of state. Perturbation of ions
is used to destroy symmetry within the supercell calculations.
We primarily use 2 × 2 × 2 (23 ) supercells (∼128 atoms) with
a -centered 23 k-point grid.
The analysis and correction methods used here are chosen
in an attempt to maximize accuracy without entailing a much
more costly analysis using more accurate electronic structure
methods, such as hybrid functionals. For fixed, experimental
lattice parameters, we have compared bulk Cu3 PSe4 calculations for GGA, GGA + U , and the Heyd-Scuseria-Ernzerhof22
(HSE) hybrid functional. We find that for a number of
properties, including the P-Se and Cu-Se bond lengths and
the lowest conduction band charge distribution, the GGA + U
results are significantly closer to the HSE results than the GGA
results. The HSE functional itself yields unexpectedly accurate
results for the band gap (error ≈ 0.02 eV) and bond lengths
(error ≈ 0.01 Å).5 The similarity of GGA + U and HSE
bulk calculations thus raises our expectations for GGA + U
performance, particularly for shallow acceptor defects which
should avoid band-gap-related uncertainties when EV = 0.
B. Defect heat of formation
The defect formation energies are performed using the
formula
HD,q (EF ,{μα }) = ED,q − EH + (EV + EF )q
μ0α + μα nα + Ecorr . (1)
+
α
The notation here follows Ref. 6: D denotes the defect type, q
is the charge of the defect charge state, EF is the Fermi energy
level, EV is the host VBM, EH is the calculated total energy
of the host supercell, and ED,q is the calculated total energy of
the defect supercell. μα ≡ μ0α + μα is the chemical potential
for atom type α in the synthesis environment with μ0α being
the calculated pure element energy (possibly with statistical
corrections10,11 ) and μα 0 being determined by synthesis
conditions. nα is the number of atoms added to the environment
in creation of the defect D.
The energy correction term Ecorr is expanded as
Ecorr = EBF + EPA + EMP
+ qEV − zh EV + ze EC .
(2)
The meaning of the last three (band-gap correction) terms
follows the description in Ref. 19 except that EV is defined
here to be positive for a gap-narrowing correction. For the
GGA + U defect calculations, we assume EV = 0, while
the correction for the GGA calculations is described in
Appendix A. EC is the correction to the conduction band
minimum (CBM), determined from EV and the experimental
and calculated band gaps of the host. For reasons discussed
below, we only apply the shallow donor correction ze EC
for the extrinsic shallow donor defects such as Zn0Cu . For all
other defects we take the “band edge only” approach to the
conduction band correction, in which we do not change the
transition energies as EC is moved. Here ze is the number of
electrons locally bound in a shallow donor state and zh is the
number of holes locally bound in a shallow acceptor state.
The first three terms of Eq. (2) are the band filling
correction, the potential alignment correction, and a modified
Makov-Payne electrostatic image correction, respectively.
These terms collectively are the finite size correction terms,
and they follow Refs. 6 and 7.
The band filling correction for the acceptor defects is given
by
EBF (D,q) = −
wk (2 − ηn,k )(ẽV − en,k )(ẽV − en,k ),
n,k
(3)
where (x) is the Heaviside step function, wk is the k-point
weight, ηn,k is the occupancy of the two-electron state (n,k),
en,k denotes the state eigenvalue, and ẽV is the host VBM
adjusted by the potential offset:
r
(4)
ẽV = EV ,H + VD,q
− VHr .
The potential references V r are calculated by averaging the
atomic sphere-averaged core potentials excluding the defect
site, and in some cases nearest neighbor sites, as described in
Ref. 7. We have not included dispersion corrections to isolated,
half occupied deep defect states, since these corrections are
found to be small in light of the much larger uncertainties of
deep state transition energies.
The potential alignment correction is given by
r
(5)
EPA = q VD,q
− VHr .
The modified Makov-Payne correction is 2/3 multiplied by
the monopole (1/L) term,
EMP =
2 q 2 αM
,
3 20 L
(6)
as derived in Ref. 6. For the 23 Cu3 PSe4 supercell, |q| = 1,
and 0 = 14.1, we find EMP = 0.069 eV.
C. Defect concentration
Defect concentration is calculated in a two step process.
The first step self consistently solves for formation temperature
concentrations n(1)
D,q of each defect type and charge state. The
second step self consistently solves for room temperature
while holding the defect
charge state concentrations n(2)
D,q
type concentrations nD = q nD,q fixed to their formation
temperature values. Nonequivalent sites of the defects VCu and
PSe have been treated as separate defects. Multiplicities include
equivalent site multiplicity and a spin degeneracy factor of two
195201-2
DEFECT PHYSICS AND ELECTRONIC PROPERTIES OF . . .
PHYSICAL REVIEW B 88, 195201 (2013)
45
40
Cu-d
Cu-d
Se-p
Se-s
total
P-s
P-p
Cu-d
Se-s
Se-p
single particle states / unit cell / eV
35
Se-p
P-p
Cu-d
30
25
Se-s
P-s
Se-s
P-s
Se-p
Cu-d
Se-p
20
Se-p
P-p
Se-p
P-s
Cu-d
15
10
5
0
-15
-10
-5
0
energy (eV)
FIG. 1. (Color online) Partial density of states for the unit cell (Cu3 PSe4 )2 . The large Cu-d peak rises to a maximum of twice the height of
the plot. The vertical dashed line at 0 denotes the valence band maximum.
for charge states with half occupied orbitals. For VCu and PSe ,
no other electronic degeneracies (or Jahn-Teller behavior) are
present. We use the full Fermi function rather than the Boltzmann approximation and calculated density of states distributions (with increased k-point density) rather than an effective
density of state approximation based on effective masses.
III. RESULTS
A. Band character
The bonding character of Cu3 PSe4 is evident in the
GGA + U calculation of the partial DOS, shown in Fig. 1.
The valence bands above −7 eV and the conduction bands
below 3.3 eV have similarities to other multinary copper
chalcogenides. One such common property is that the Cu-d
states are split into nonbonding e orbitals and t2 orbitals which
form filled bonding and filled antibonding bands because
of their interaction with the chalcogenide p orbitals.23 The
antibonding band forms the highest valence band. Like
CuInSe2 , CuGaSe2 , and Cu2 ZnSnSe4 ,23 the conduction band
has a character that is largely antibonding between Se-p and
Mt-s, where Mt represents the element acting as the high
valence metal (e.g., Sn in Cu2 ZnSnSe4 , P in Cu3 PSe4 ). The
antibonding character is inferred from the presence of a spatial
node between the Mt and Se atoms in the charge density of
the lowest conduction band.5 Unlike materials with a metallic
Mt, Cu3 PSe4 has no valence band that is the obvious bonding
counterpart. In fact, the P-s orbitals have nominally been
filled in the P-s/Se-s bonding and antibonding bands, near
−15 and −10 eV. This σ σ bonding does not occur when
Mt is more metallic, because of the larger energy difference
between the atomic Mt-s level and the chalcogenide s level.
Thus the appearance of a P-s/Se-p ∗ antibond is somewhat
surprising despite the fact that it follows the trend of other
multinary copper chalcogenides. The bonding counterpart of
the second conduction band, which has significant P-p/Se-p∗
character, is found in the valence band near −5.7 eV.
B. Effective mass and dielectric properties
The calculated GGA + U effective hole mass and dielectric
tensor components are shown in Table I. The dielectric tensor is
calculated using density functional perturbation theory.24 The
effective mass tensor, calculated from the band structure, has
much larger components in the yz plane than along the x axis.
Because the radius of a hydrogenic shallow defect state (also
known as a perturbed host state6 ) is inversely proportional
to effective mass, this results in the shallow acceptor VCu
wave function being greatly elongated in the x direction.
The conductivity effective hole mass is m∗cond ≡ 3/ i m−1
i =
0.27 m0 . For comparison, the Si light and heavy hole effective
masses are 0.16 and 0.49 m0 , respectively.
C. Chemical potential domain
We analyze the allowed chemical potential domain for
Cu3 PSe4 synthesis by calculating H for 22 compounds
TABLE I. Principal axis tensor components and appropriate
scalar averages for effective hole mass (units of electron mass m0 )
and electronic and total dielectric constants, ∞ and 0 .
x
m
(elect.)
(total)
195201-3
0.10
14.0
16.8
y
1.66
13.1
14.8
z
scalar
1.82
12.0
13.6
m = 0.67, m∗cond = 0.27
∞ = 13.0
0 = 15.1
∗
FOSTER, BARRAS, VIELMA, AND SCHNEIDER
ΔμP (eV)
-1.4
-1.2
-1
-0.8
-0.6
-0.4
P,Cu rich
0
0
-0.2
Cu7PSe6
-0.2
Cu3PSe4
-0.4
ΔμCu (eV)
CuSe
Cu3Se2
ΔμSe = 0
Se rich
SeP
2
PSe
-0.6
Cu poor
FIG. 2. Chemical potential domain with stable region of Cu3 PSe4
in gray. The chosen Cu-rich growth condition is indicated by a circle.
Formation energy ΔHD (eV)
P poor
-1.8 -1.6
PHYSICAL REVIEW B 88, 195201 (2013)
CuSe
1.5
1
ZnCu
VSe
D. Defect analysis
The defect analysis is performed initially using a 23
supercell (128 atoms). We use all finite size corrections
described above (EBF , EPA , EMP ). For the GGA + U calculation, no correction is made to the valence band, while the
conduction band correction EC = 0.88 eV is obtained from
the difference of the experimental band gap (1.4 eV) and the
calculated band gap (0.52 eV). A shallow donor correction
term is applied to the energies of incompletely ionized shallow
donor defects. However, none of the intrinsic point defects are
clearly shallow donors, and thus this correction is applied only
for the extrinsic donors considered: Ca, Cd, and Zn on a Cu
site, and Cl on a Se site. [Here the correction is +(1 − q)EC ,
since ze = 1 − q with q = 0,1.]
CuP
0.5
EF,SC
containing Cu, P, and Se. Figure 2 shows the results for
several important compounds, revealing a relatively large
stable chemical potential domain. To best match experimental
carrier concentrations,8 we perform the defect calculations
for the conditions μP = 0 and μCu = −0.11 eV (circled
in Fig. 2). Choosing μCu to assume its maximum allowed
value minimizes the calculated concentration of the shallow
acceptor defect VCu .
We note that it has been observed25 that under certain
conditions Cu3 PSe4 can coexist with the ionic conductor
Cu7 PSe6 , but this is not predicted by chemical potential
domain analysis. This discrepancy may be due to finite
temperature effects; the low temperature α phase26 of Cu7 PSe6
was used in calculations, while at formation temperature
the partially disordered γ phase would be present. We also
note here that the error of total energy calculations involving
phosphorus can be large; a statistical correction of 0.6 eV per
P atom is given in Ref. 10 due to artefactual energy differences
between phosphorus in reductive and neutral (elemental)
environments. This error is expected to impart uncertainty
both to the calculated heat of formation of Cu3 PSe4 , which
affects defect energies through its effect on μCu , and to the
defect supercell energies themselves, particularly for the high
concentration PSe defect. In the latter case, the additional P
atom is reduced by the neighboring Cu ions, in strong contrast
to the host P atoms, which are oxidized by the Se neighbors.
While phosphorus raises concern, the GGA + U statistical
corrections10 associated with Cu and Se atoms are less than
0.05 eV, and our calculated heat of formation of Cu3 Se2 is
within 0.05 eV of experiment.10
VP
VCu
PSe
EC,calc
0
0
0.2
0.4
0.6
0.8
EF (eV)
1
1.2
1.4
FIG. 3. Defect formation energies and transition energies. Where
nonequivalent sites are calculated, the lowest energies for each charge
state are shown. The self-consistent room temperature Fermi energy
EF,SC , assuming a formation temperature of 500 ◦ C, is shown by the
vertical line at 0.031 eV. The (0/ + ) transition energy for shallow
donor ZnCu has been raised to follow the conduction band correction.
Formation energies and transition energies for the lower
energy intrinsic defects and the lowest energy extrinsic defect
are shown in Fig. 3. The acceptors VCu and PSe both pin the
Fermi energy below mid-gap, preventing Cu3 PSe4 from being
n doped near thermal equilibrium. The formation energy of
the neutral defect V0Cu is calculated to be 0.50 eV, with a
(−/0) transition energy of 0.05 to 0.06 eV, depending on the
Cu site. The formation energy of P0Se varies with site from
0.47 to 0.50 eV, with the (−/0) transition energies varying
from 0.08 to 0.17 eV.
The net finite size corrections for the two important intrinsic
defects VCu and PSe lie between 0 and 0.11 eV, with the
potential alignment correction ranging from 0 to 0.06 eV.
For these defects, only the charge neutral defect states have
nonzero EBF values (0 to −0.04 eV). A band filling correction analogous to Eq. (3) is used for the extrinsic donor defect
ZnCu , yielding EBF = −0.08 eV for the neutral defect state.
We use the formation temperature of 500 ◦ C (approximately
the temperature used in recent pellet and single crystal
experiments8 ) to calculate the concentrations of each defect
type. The resulting defect concentrations (irrespective of
charge state) are 4.1 × 1019 cm−3 for VCu and 4.6 × 1019 cm−3
for PSe . The second step of the concentration calculation
yields a room temperature (300 K) self-consistent Fermi level
of 0.031 eV above the VBM and a hole concentration of
p = 8 × 1018 cm−3 . VCu is electronically the most important
defect type, since the contribution of V−
Cu to the hole density
is over five times that of P−
Se .
If zinc is present during synthesis, the maximum Zn+
Cu
concentration is approximately 5 × 1018 cm−3 , and the net
room temperature hole density is lowered slightly to p =
6 × 1018 cm−3 . The other potential donor dopants considered
have greater formation energies and can be neglected for all
growth conditions.
We have recalculated the charged configurations of the
weakly localized VCu defect using a 43 (1024 atom) supercell.
195201-4
DEFECT PHYSICS AND ELECTRONIC PROPERTIES OF . . .
PHYSICAL REVIEW B 88, 195201 (2013)
Even for this supercell size, the defect wave function is not
localized within the supercell in the x direction (the low
effective mass direction).
For the neutral defect the calculated (−/0) charge transition
energy is 0.04 eV and H = 0.53 eV. We note that the
hydrogenlike approximation using the conductivity effective
mass yields a comparable binding energy of 0.02 eV. Assigning
the large supercell data to all Cu sites, in combination with
the previous PSe data, yields an insignificantly modified hole
density p = 9 × 1018 cm−3 .
The PSe defect state is substantially localized within the
smaller 23 supercell. The defect state has P-p character on the
defect (Se) site and Cu-d character on the nearest neighbors,
similar to a localized version of the host valence band, which
has Se-p and Cu-d character. The degree of localization allows
us to apply the defect image charge correction of Ref. 9 using
the neutral defect potential as the reference potential (see
Appendix B). The resulting correction (0.08 eV) agrees
well with the corresponding correction (EMP + EPA =
0.07 eV) according to Refs. 6 and 7.
E. Discussion and further investigation
The defect analysis performed here agrees qualitatively
with recent experimental results. Our calculated Cu:P ratio
of 2.97 is consistent with the value 2.92 ± 0.06 measured
for single crystals.8 We predict a large hole concentration
of p = 8 × 1018 cm−3 , about one order of magnitude larger
than the value 6 × 1017 cm−3 obtained by Hall and Seebeck
measurements on pressed, sintered pellets.8
We compare the GGA + U defect calculations described
above with standard GGA defect calculations followed by
application of a GGA + U correction6,19 (−0.34 eV) to the
VBM. The GGA defect calculations include all types of
corrections applied to the GGA + U calculations and include a
GGA determination of the maximum allowed copper chemical
potential μCu (−0.06 eV). As shown in Table II, the more
standard “GGA + VBM correction” procedure changes the
formation energies of V0Cu and V−
Cu (evaluated at maximum
μCu and minimum EF ) by about −0.16 eV, causing a
significantly larger overestimation of p relative to reported
experimental values. This comparison shows that GGA + U
performs better than GGA not only in bulk total energy
calculations,10,11 but also in defect calculations.12
We consider possible errors resulting from using a single
effective formation temperature of 500 ◦ C to establish total
defect concentrations. Ionic diffusion at room temperature is
highly unlikely to be significant on any realized time scale.27
However ionic diffusion during cooling, or overestimation of
the formation temperature, which is not uncommon,28 may
result in lower calculated defect and carrier concentrations.
For effective formation temperatures of 450 ◦ C and 400 ◦ C, the
self-consistent hole concentration p is modestly lowered (from
7.8 × 1018 cm−3 ) to 6.2 × 1018 cm−3 and 4.8 × 1018 cm−3 ,
respectively. Thus, uncertainties in the effective growth
temperature can explain some of the differences between
calculated and experimentally reported hole concentrations.
It is instructive to consider further the implications of
the available experimental results.8 We examine possible
changes in defect formation enthalpies which would bring the
calculated hole concentration p closer to the value measured
for polycrystalline pellets.8 If one assumes that the calculated
transition energy of VCu is not underestimated, the experiments
of Ref. 8 indicate that the formation energy of VCu must
increase, while the transition energy of PSe increases and
the formation energy of PSe decreases. The adjustment to
the VCu energy must be significant to recover the measured
p. For example, increasing the formation energy of VCu
defects by 0.35 eV while applying changes of −0.05 and
0.05 eV to the neutral and charged PSe defects respectively
yields p = 7 × 1017 cm−3 and a Cu:P ratio of 2.96. Such
q
large changes to the VCu formation energies cannot readily
be explained by systematic calculational errors associated
primarily with phosphorus.
An alternative possibility is that the GGA + U calculated
VBM is too high by a modest amount, and that the apparent
shallow character of the VCu defect is an artifact of this
band misplacement. For example, applying a valence band
correction EV = −0.1 eV and choosing not to apply the
shallow acceptor corrections to the neutral defects (that is,
using a strictly “band edge only” approach) yields the much
lower hole concentration p = 1.1 × 1018 cm−3 with an only
slightly increased Cu:P ratio (2.976).
The experimental data suggests an increase in neutral P
concentration, and possibly the presence of a low energy
donor defect involving extra P atoms. Such a donor defect
could lower the hole concentration by compensating the
VCu acceptors and thus avoid the need to raise H (VCu ).
We therefore have examined, at lower accuracy and without
finite size corrections, a number of neutral and positively
charged P-rich defects, including interstitials and complexes
in various configurations. The results are shown in Fig. 4. The
energies suggest that there are no significant sources of extra
phosphorus besides PSe .
ΔH(EF = 0) for various P-rich defects
q=0
q = +1
IP-IP*
IP-PSe*
TABLE II. Calculated site-averaged formation energies for VCu
defects and predicted versus experimental hole concentrations. The
error of the GGA + U method is seen to be smaller than the more
standard6 method of using GGA including VBM corrections.
GGA + VBM corr.
GGA + U
Hall measurement (Ref. 8)
H (V0Cu ) [eV]
H (V−
Cu )
p [cm−3 ]
0.34
0.50
0.40
0.56
6 × 1019
8 × 1018
6 × 1017
PSe-PSe*
IP*
PSe
* no corrections
0
1
2
3
energy (eV)
4
5
FIG. 4. Heat of formation (GGA + U ) of interstitial and complex
defects containing extra P atoms. Energy corrections Ecorr are
neglected except for the PSe defect.
195201-5
FOSTER, BARRAS, VIELMA, AND SCHNEIDER
PHYSICAL REVIEW B 88, 195201 (2013)
IV. CONCLUSION
In conclusion, we have performed a set of GGA + U defect
calculations on Cu3 PSe4 , a p-type semiconductor with a direct
band gap of 1.4 eV. We compare our methods against standard
GGA, larger supercells, and alternative correction methods.
We predict that the VCu defect is mostly responsible for the
large, experimentally observed intrinsic hole concentration p,
with some contribution from PSe . Both of these defects pin the
Fermi level below mid-gap, so that n doping is prohibited
near thermal equilibrium. Both defects also contribute to
the observed nonstoichiometric Cu:P ratio. Our calculation
overestimates the hole concentration p by about one order
of magnitude. Overall, the GGA + U method is shown to be
more accurate than standard GGA calculations with valence
band corrections. Doping with Zn is calculated to have a
small but noticeable effect on p. Because of the apparent
uncertainty in the calculations however, this analysis does
not rule out the possibility that Zn doping could significantly
reduce p.
ACKNOWLEDGMENTS
We thank Robert Kokenyesi and Janet Tate from Oregon
State University (OSU) for helpful discussions. We thank
Stephan Lany from the National Renewable Energy Laboratory for very helpful discussions and for providing scripts
with which some of the potential alignment and band filling
corrections are calculated. This work has been supported by the
National Science Foundation of the USA under Grant SOLAR
DMS-1035513.
APPENDIX A: THE GGA + VBM CORRECTION
CALCULATIONS
Except for the final VBM correction, the chemical potential
domain analysis and defect analysis for the GGA calculations
are performed with U = 0. Similar to the set of GGA + U
calculations, bulk relaxations are performed for the unit cells
of elements Cu, P, Se, and relevant compounds such as Cu3 Se2
and PSe, in order to obtain the chemical potential domain and
determine the maximum μCu . Relaxed lattice parameters
for Cu3 PSe4 are also recalculated with U = 0 in order to
create the 23 supercell for the GGA defect calculations. Defect
calculations are performed with the same types of corrections
as are used for the wholly GGA + U method.
*
Guenter.Schneider@physics.oregonstate.edu
L. Yu and A. Zunger, Phys. Rev. Lett. 108, 068701 (2012).
2
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1
The value of EV is determined in the following manner.
A static (ion-fixed) GGA + U calculation of bulk Cu3 PSe4 is
performed using the GGA-relaxed unit cell. The energy of
the resulting VBM relative to the mean energy of the Se-s
peak (used as a reference) is taken to be the relative VBM
of the GGA + VBM correction method. This energy, minus
the corresponding relative VBM of the plain GGA unit cell
calculation, gives the valence band correction EV :
GGA + VBM corr.
EV = EVGGA + VBM corr. − ESe-s
GGA
.
(A1)
− EVGGA − ESe-s
We find that an alternate reference, the average electrostatic
potential of spheres centered on the Se atoms, results in
negligible differences.
APPENDIX B: ALTERNATIVE “MODEL CHARGE”
ELECTROSTATIC CORRECTIONS
We performed the alternative electrostatic + potential alignment correction as described in Ref. 9, with the exception that
the electrostatic potential from the neutral defect calculation
was used as a reference potential, instead of the potential of
the host supercell. This was necessary in order to locate the
potential asymptote away from the defect. When the host
potential was used as a reference, the motion of the ions
upon relaxation caused extreme oscillations in the potential
difference VD,q − VH . The electrostatic potential difference
VD,q − VD,0 on the other hand, involved much less radical
oscillations due to the relative ionic motion, and allowed the
asymptote to be located. (This issue was avoided in Ref. 9 by
not allowing ionic relaxation.)
Operationally, we constructed a model, periodic, spherical
Gaussian + exponential charge distribution on the same real
space lattice that was assumed by the DFT/PAW charge and
potential distributions. Periodicity was realized through the
Fourier representation of the functions. The Gaussian width
parameter β was set to 2 bohr, as done in Ref. 9. The ratio x
of exponential to Gaussian character was parametrized by a
periodic variable t. The parameter t, the exponential width parameter γ , and the center of the charge distribution (x0 ,y0 ,z0 )
were fit to the defect charge state using a constraint on min(γ ).
The program SXDEFECTALIGN by Christoph Freysoldt was then
used to obtain the final corrections.9
7
S. Lany and A. Zunger, Model. Simul. Mater. Sci. Eng. 17, 084002
(2009).
8
V. Itthibenchapong, R. S. Kokenyesi, A. J. Ritenour, L. N. Zakharov,
S. W. Boettcher, J. F. Wager, and D. A. Keszler, J. Mater. Chem. C
1, 657 (2013).
9
C. Freysoldt, J. Neugebauer, and C. G. Van de Walle, Phys. Status
Solidi (b) 248, 1067 (2011); Phys. Rev. Lett. 102, 016402 (2009).
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S. Lany, Phys. Rev. B 78, 245207 (2008).
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115104 (2012).
12
D. O. Scanlon, B. J. Morgan, and G. W. Watson, J. Chem. Phys.
131, 124703 (2009).
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DEFECT PHYSICS AND ELECTRONIC PROPERTIES OF . . .
13
PHYSICAL REVIEW B 88, 195201 (2013)
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When P is oxidized, an appropriate correction would likely have
the opposite sign. Thus setting the correction to zero in these cases
is a conservative measure.
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20
Effective U values including 4 eV (Ref. 23), 5 eV (Ref. 10), and
6 eV (Ref. 19) have been used for Cu-d to fit the photoemission
experiment.
21
While the present GGA + U method is unlikely to place the
VBM with an accuracy better than 0.1 eV, it is difficult to obtain
such accuracy by any other standard method, including the GW
approximation. We note that a GGA + U + G0 W0 calculation
using 288 bands (62 filled) at experimental lattice and ionic
parameters yields a slight decrease (−0.01 eV) in the VBM as
compared to GGA + U .
22
J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118,
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Y. Zhang, X. Yuan, X. Sun, B. C. Shih, P. Zhang, and W. Zhang,
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X. Gonze and C. Lee, Phys. Rev. B 55, 10355 (1997).
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R. Kokenyesi (private communication).
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27
Cu3 PSe4 has a tetrahedral microstructure (Ref. 5) found in typical
wurtzite and zinc-blende semiconductors, and is structurally very
unlike the stoichiometrically similar ionic conductor Cu7 PSe6 ,
which has chains of neighboring Cu ions.
28
J. Tate (private communication).
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